Device for the transformation of mathematical functions



J 1946. P. J. s. ENGER ET AL 2,402,321

DEVICE FOR THE TRANSFORMATION OF MATHEMATICAL FUNCTIONS Filed July 16,1945 4 Sheets-Sheet 1 A'TTYS.

June 18, 1946.

P. J. S. ENGER ET AL 2,402,321 DEVICE FOR THE TRANSFORMATION OFMATHEMATICAL FUNCTIONS Filed July 16, 1943 4 Sheets-Sheet 2 \IIIIIII I,llllllllll IIIIIIIII A II El" XNVENTQQS J. SENCTER RFARsm-um June 18,1946. P. J. s. ENGER ET AL DEVICE FOR THE TRANSFORMATION OF MATHEMATICALFUNCTIONS Filed July 16, 1943 4 Sheets-Sheet 5 TNVENT R .ISTLNGER A.PARscHxu June 18, 1946. P. J. s. ENGER ETAL 2,402,321

DEVICE FOR THE TRANSFORMATION OF MATHEMATICAL FUNCTIONS Filed July 16,1943 4 Sheets-Sheet 4 ATTYS.

Patented June 18, 1946 DEVICE FOR THE TRANSFORMATION OF MATHEMATICALFUNCTIONS Per Johan Samuel Enger and Alexander Parschin,

Stockholm, Sweden, assignors to Telefonaktiebolaget L. M. Ericsson,Stockholm, Sweden, a

company of Sweden Application July 16, 1943, Serial No. 495,026 InSweden June 25, 1942 The present invention relates to a device fortransformation of mathematical functions. It refers particularly to suchcalculating machines wherein mathematical functions with two or severalvariables are reproduced in the shape of curve bodies of a mechanical,electrical or other nature,

in such a way that certain variables are represented by the movements ofa feeler in a certain co-ordinate system related to the curve body,whilst one variable is represented by the deflexion of a mechanical,electrical or other kind which the curve body produces in the feeler.The invention is applicable when the mathematical functions which arereproduced are symmetric or may be transformed into a symmetric form.

The object of the invention is to reduce the dimensions of the curvebody without reducing the accuracy of feeling of the variablesrepresented by the curve body. This is obtained by making the curve bodyreproduce only the ranch of the symmetric function which lies on oneside of the point of symmetry. Should the incoming variable pass overits entire range, the feeler first passes once over the curve body whenone branch has been passed over. At the point of symmetry the feelerturns and passes over the curve body a second time in the oppositedirection when the second branch is passed over. The device according toinvention should therefore be designed preferably so that a drivingdevice, the position of which represents the instant value of thefunction to be transformed, determines the position of a driven device,the position of which represents the instant value of the transformedfunctionwhereby the driving device consists of two parts connectedtogether in such a way that, one part moving in one direction, the otherpart moves in the opposite direction, the driven device being arrangedto engage upon the part of the driving device which for the momentoccupies the most advanced position in a certain direction.

5 Claims a driven shaft in an arrangement for the transformation offunctions.

Fig. 6 is a graphic picture of the relation between the movements of thedriving and the driven shaft in Fig. 5.

According to Fig. l, where the mathematical function is linearlysymmetric, the two branches of thefunction are inversely congruent.Origin of co-ordinate axes has been placed in the point of symmetry.Here is The deflection y of a feeler indicates the value of the functionon both sides of the line of symmetry. If however the mathematicalfunction is polarly symmetric so that the two branches are directlycongruent, as in Fig. 2, where the feeler marks the numerical value ofthe function but its sign must besides be indicated in a suitablemanner.

The components of a device for transformation of functions areillustrated by the example as shown in Fig. 3.

A driving-shaft H, which supports a driving arm 12, moves concentricallywithin a hollow shaft 13, which supports a driving arm [4. A gear l5,l6, H is 50 arranged that, when the shaft I I turns the driving arm l2in one direction. the shaft 13 turns the driving arm l4 equally as muchin the other direction. The turning angle 0: of the driving shaft l l isa representation of the incoming variable of the mathematical function.An arm l8, supported by a shaft I9, is by a spring 20 pressed againstthe driving arms 12 resp, 14. If the driving shaft II is turned from theposition shown in one direction, for instance clockwise, the driving arml2 moves the arm l8 in the direction of the driving shaft. If, on theother hand, the driving shaft is turned from the same starting positionin the opposite direction, the driving arm l2 moves the arm I8 at firstin the same direction as the shaft ll, i. e. counterclockwise. When theshaft H ha been turned half the angular distance between the drivingarms I2 and M, the driving arm 14 will however move the arm l8 in adirection opposite that of the driving shaft i. e. clockwise. Themovement which the arm l8 receives from the driving arms is by the gearwheels 2| and 22 transferred to an arm 23, which supports aslidingcontact 24, which moves along a contact row 25. The turning angle of thesliding contact=fi. An electric current source is supposed to besuitably connected to the aacasei contact row 25 so that the potentialof each individual contact point becomes a certain, desired function ofthe respective position of the contact point in the contact row. Thesliding contact 24 is electrically connected to a receiver 26, and inthis way the potential of the contact point on which the sliding contact24 stands is transferred to the receiver 26. Thus the relationshipbetween the movements of the sliding contact as and driving shaft II isdependent on the position of the shaft I9 compared to the shafts II andIs.

At first, it is assumed, for the sake of simplicity, that the shafts II,I3 and I9 are concentric. The turning angle of the arm is is then equalto the turning angle of the driving arm (52 or ii) against which it ispressed for the moment. In the position where the driving arms i2 and Il pass each other, i. e. for u=0, both arm touch i8 which is justoccupying an extreme position in the counter-clockwise direction. Theposition 3:0 held by the sliding contact 2Q along the contact row isalso an extreme position and represents the symmetrical value of thefunction illustrated.

If from the symmetric value as a starting-point the driving shaft I I isturned a certain a'rbitrary angle the sliding contact 24 will be movedcounter-clockwise an angle 5, which is proportionate to the numericalvalue of the turning angle a of the driving shaft II, regardless ofwhether the turning of the driving shaft goes clockwise orcounter-clockwise. The electric voltage which the contact row conveys tothe sliding contact is a function of the angular position ,8 of thesliding contact reckoned from the symmetrical point and becomestherefore a linearly symmetric function of m, the symmetric value (1:0.

If the function to be represented is linearly symmetric the tension ofthe sliding contact represents the value of the function. Should howeverthe function to be represented be polarly symmetric the tension of thesliding contact reppresents the numerical value of the function.Therefore the sign of the function must also be indicated which may bedone in the following way:

The positive pole of a current source 28 is connected to the drivingarms It. The arm I8 is connected to a receiver 21 which is alsoconnected to the negative pole of the current source. The receiver 2'!is consequently traversed by a current as soon as the arm I8 touches thedriving arm I 4i. Should the arm I8 only touch the driving arm I2, thecurrent flowing through the receiver 211 is interrupted.

The device described may be used (A) as sender (a) of a polarlysymmetric function, (h) of a linearly symmetric function, and (B)asreceiver of a polarly symmetric function.

Theseways of application will now be described.

(A) Sender The variable a: contained in the function y=f(.r) isrepresented by the mechanical movement of the driving shaft I i. Thismovement may be obtained in various ways, for instance through manualsetting or by connecting an apparatus which gives the variable a: in theshape of a mechanical movement to the driving shaft, either mechanicallyor by means-of a system of trailing scales or by means of a servo-motor.

A certain value x corresponds to a certain position of the driving shaftII and thus also to a certain position of the sliding contact. Thus acertain potential will pertain to each value :c which potential is sentto the receiver 26. This potential represents the value y correspondingto the value a: set for the moment.

(a) For the polarly symmetric function y=f(:c) applies f(a:)=-f(.l:)..

The branch of the function which lies on on side of the symmetric valuethus numerically equals the branch lying on the opposite side of thesymmetric value but it has the opposite sign. As mentioned above thepotential which is conveyed from the sliding contact to the receiver 26illustrates the numerical value of the function. The change of signswhich takes place in the function as the symmetric value is passed willbe indicated by the receiver 2'! being tie-energized On one side of thesymmetric value but traversed by a current on the other side of thesymmetric value.

(b) For the linearly symmetric function y=,f(a:) applies ,f(.r)=f(a:).

Here the branches of the function on each side of the symmetric valueare identical, thus rendering the receiver 2? and its electricconnection superfluous.

(B) Receiver of a. polarly symmetric function In the Fig. 4 the partsI03 and I04 illustrate a sender arrangement not included in theinvention proper which arrangement contains two devices, the one I03emitting a voltage which represents the numeric value [I'll of thevariable y, the second I04 emitting a voltage representing the sign ofthe variably y. The invention comprises two relay devices I26, I21, oneof which I26 receives the valtage, which represents I21], and the otherI 21 receives 'the voltage representing the sign of the variable 11.Below, to the right, an arrangement i shown which emits the variable win the shape of a mechanical movement.

To make the diagram in Fig. 4 more easily grasped, another embodiment ofthe invention than the one shown in Fig. 3 has been used, the contactrow I25 being linear and the driving arms I I2 and H4 with their gearwheels having the shape of straight racks, one of which being for themoment placed topmost and arranged to lift the sliding contact I24 whichis pressed down by the force of gravitation. The receivers I26 and i2?control a servo-motor I 3I I32, which drives a driving shaft H16 whichover a gear device III actuates the driving arms I I2 and III. Animpulse to the coil I 3| resp. coil I32 in the servomotor turns thedriving shaft I06 clockwise resp.

' counter-clockwise to lower resp. higher values x1, thus causing thesliding contact I24 to move in a corresponding manner and its voltage111 to change according to the function y=f(a:). The electric contactbetween the driving arm H4 and its arm is shown as a contact I33 whichis broken for positive values 1/, i. e. when the driving arm I I2 has aposition higher than the driving arm H4, and closed for 215 i. e. whenthe driving arm I is on the same height or higher than the driving armH2. The driving arm I I2 is supposed to actuate a pointer I34 whichpoints to a scale I35 on which the value a: may

.be read.

The receiver I26 is a polarized relay, the coil I61! of which isconnected partly to the incoming voltage [1 and partly to the slidingcontact I24, the voltag of which may be called M1. The coil actuates acontact device i6I, I62, the movable contact spring of which isconnected to the pole of a power source. Ii the voltage of the slidingcontact I2! is less resp. higher than the incoming tension, the contactI6I respectively I82 closes; If the two voltages are identical bothcontacts are broken. The receiver I2I consists of two relays III andI8I. The coil of the rela I8! is connected to the receiver I04 and thearmature of the relay is thus in rest position if y 0, but in operatingposition if 11:0. The coil in the relay III is connected to the contactI33 and the relay armature is thus in rest position respectivelyoperating position according to the position of the driving arms H2 andIll corresponding to y1 O respectively 11150. The operating manner is asfollows:

(I) For values a: 0. The sender I05 does not emit current, wherefore thearmature of relay I8I remains in rest position.

(1) The receiver has the value :m 0. The sliding contact is supported bythe driving arm II2, the con tact I33 is broken and the arma= of relayIII has rest position. If the voltage lyh of the sliding contact I24 hasa higher respectively lower value than the tension lyl originating fromthe send er I03, viz,

current passes through the relay coil I60 in the directionof re=spectively from the sliding conoontact I24. The contact H52 respectivelI6I is then closed and the contacts I02--II3 respective ly I6I-i82 areenergized, whereby the servo-motor I 3! respectively I32 drives theshaft I06 clockwise respectively counterclockwise. The driving arm H2 islowered respectively raised, whereby the voltage lyli of the slidingcontact I24 falls respectively rises, up to |y|1=|yI. The relay I60 thenbecomes deenersized, the contact I62 respectively I6I is broken and theservo-motor stops. The receiver has then the position 331:3),corresponding to yi=zl=f(:r).

(2) The receiver has a value auil. The

contact is supported by the driving arm III, the contact I33 is closedand the armature of relay I'II has operating position. Hegardless of thevoltage of the sliding contact H4 and the condition of the relay I60 9.current passes from the contact spring I85 connected with the pole ofthe current source, over I14, I12, to the coil I32, whereby theservomotor I3I, I32 drives the shaft I08 counter-clockwise. The drivingarm III is lowered. When the sliding contact has reached the extremeposition I29 it is lifted by the driving arm H2 and leaves the drivingarm Ill. At the same time the contact I33 is broken, the relay I'llfalls and the procedure continues according to the case (I1) until thecorrect on value, mi=zc has been reached.

6 (II) For values a o. The sender I04 emits a current to the relay I8Iwhich lies with its armature energized.

(1) The receiver has the value mro. The sliding contact is supported bythe driving arm I, the contact I33 is closed, the armature of relay IIIis energized. If the voltage 1 of the sliding contact has a higherrespectively lower value than the voltage lyl deriving from the senderI03, viz. lyli lylr lyl, a current passes through the relay coil I60 inthe direction from respectively to the sliding contact I24. Then thecontact I62 respectively IGI is closed and current passes over thedevices I62I I2-I 32 respective ly I6II83I3I, whereby the servo-motori32 respectively I35 drives the shaft I06 counterclockwise respectivelyclockwise. The driving arm H41 i lowered respectively raised whereby thevoltage lyli of the sliding contact i24 decreases respectively increasestill ]y[1=|z/|. The relay E60 now is de-energized, the contact I62respectively IN is bro ken and the servo-motor stops. Th receiver hasthen a position :cr=:c corresponding to (2) The receiver has the value:r |'l. The sliding contact is supported by the driving arm M2, thecontact I33 is broken and the relay IT! is de-energized. Irrespective ofthe voltage of the sliding contact I24 and the condition of relay I60 acurrent passes from the contact spring I over the devices I15, I13 tothe contact I3I, whereby the servo-motor drives the shaft I06 clockwise.The driving arm H2 is lowered. When the sliding contact has reached theextreme position I29 it is lifted by the driving arm I I4.Simultaneously the contact I33 is closed, relay I'II attracts and theprocedure continues according to the case (I11), until the correct values ai -0:, has been attained.

A closer description of the relationship between the movements of thesliding contact 24 and those of the shaft II will now be given on thebasis of Fig. 5 which shows a modified detail of the device according toFig. 3.

The driving shaft II which supports the driving arm l2 movesconcentrically within the shaft I3 as shown in Fig. 3. The shaft I3carrying the driving arm I4 is however not represented in Fig. 5. Theshafts I9 and II in Fig. 5 are, contrary to the corresponding shafts inFig. 3, not concentric but situated at a distance 0 from each other. Thearm I8 with the length a is placed on the shaft I9, as is also the gearwheel 2I which drives the gear wheel 22 which supports the arm with thesliding contact 24. The sliding contact runs. along the contact row 25which is composed of a great number of close-lyin and uniformly spacedmetallic contacts.

The arm i8 supports at'its free end a roller 39, A spring which is notshown in Fig. 5 tends to turn the arm it so that the roller 80 ispressed against the driving arm as. The end point of the arm I8 willthus continuously be on a straight line parallel to the active edge ofthe driving arm. The space between this line and the driving shaft 8 iis indicated by d.

In the drawings, A signifies the angle'between the active edge of thedriving arm i2 and the con- 50 necting line between the shafts ii and39. B is the angle between arm it and the prolongation of the saidconnecting line. K represents the position of the sliding contact 24reckoned from the position it holds when 5:0, viz. when the arm it liesin the prolongation of the connecting line between the shafts ii and i9.

r1=the pitch radius oi the gear wheel 2 i rz=the pitch radius of thegear wheel 22,

k1=the number of contact steps per angular unity 29 of K and n=the totalnumber of contact steps reckoned from the extreme position.

Now it is d a sin A sin A sin (B-A) n=k K=k B In Fig. 6 is shown as afunction of A for diflerent values of old, when (i=0.

For

viz. when the shaft i9 is concentric to the shaft i i, a constantrelation exists K/ A. when 6 1 E 0 I the relation K/A has its highestvalue for A=o.

When A increases K/A decreases first slowly and then quickly. For

' ent value for difierent values of A. The angular change in Acorresponding to the progress of a contact step on the contact row thusvaries with A.

If the arrangement is used as a receiver of a function which isillustrated in the shape of an electric voltage and the contact row hasthe shape of equally spaced, separate contact points, an adjustmentcannot take place with a greater ac-.- curacy than half a contact space.If the right potential for which the apparatus is to be adjusted, liesbetween the potentials of two adjacent metallic contacts, the slidingcontact may attain the potential of one or the other of these metalliccontacts by touching one of them. Possibly, by touching bothsimultaneously, it may attain a certain potential which lies somewherebetween the potentials of both metallic contacts, but any is a 3 furtheradjustment to the right potential is not possible.

Since the accuracy of adjustment ofthe sliding contact is limited by thesize of the contact step, the adjustment accuracy of the driving shaftwill thus be limited to the angle which corresponds to the movement ofone contact step of the sliding contact. The accuracy of the adjustmentof the contact arm 28 is constant and equals one (or possibly a half)contact step. The relation in the adjustment of the contact arm 26 andthe driving arm 82 fluctuates with A and consequently the accuracy ofadjustment for the driving arm will vary with A. A is thus set withvarious accuracy within different parts of its held.

if a limited number of metallic contacts may be disposed of, it mayoccur that the variable to be illustrated embraces such a large rangethat the accuracy which is obtained at the setting to within one contactstep does not suflice if the accuracy of adjustment [be made constantover the entire range. The transformation of the variable oflered by thedescribed arrangement may then be used to ensure a greater accuracywithin a part of the range by reducing the accuracy within other partsof the range.

The transformation may be adapted to the requirements placed on thedistribution of accuracy by the possibility firstly to adapt therelation c/o, secondly to choose suitable limits for the movement of thedriving arms (1. e. Amam A Amim and simultaneously to adapt the gearingratio to a suitable value.

If the mathematical motion is of a nature admitting the regrouping ofthe accuracy, the arrangement offers a gain because, if it be tried toillustrate the function without making use of the possibility oftransformation, it would be necessary to maintain the required maximumaccuracy over the entire range which would entail that the number ofmetallic contacts must be larger than when using a fluctuating accuracy.

The object of the invention may naturally be gained by using other meansoi executing the transformation wherefore the invention is not limitedsolely to the arrangement described.

We claim:

1. An apparatus of the character described comprising a driving device,a driven device, the position of the driving device representing theinstant value of the function to be transformed, and the position of thedriven device determining the instant value of the function transformed,said driving device including two relatively movable members and meansconnecting the said members whereby they are caused to move in oppositedirections, said driven device including a member engaged and operatedby one of said two relatively movable members which at the momentoccupies the most advanced position in a certain direction.

2. An apparatus as claimed in claim 1 in which the driving device alsoincludes two concentrically arranged shafts rotatable in oppositedirections and supporting said relatively movable members.

3. An apparatus as claimed in claim 1 wherein said relatively movablemembers are constituted by racks moved in opposite directions by saidconnecting means.

4. An apparatus as claimed in claim 1 wherein said relatively movablemembers are constituted 10 by arms supported on concentrically arrangedshafts and wherein. the last mentioned member is constituted by an armmounted on a shaft disposed excentrically with respect to the first men-5 tioned shafts.

PER JDHAN SAMUEL ENGER. AmANDER ransom.

